Abstract: A scheme of codimension is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous matrix and is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers and we denote by (resp. ) the locus of good (resp. standard) determinantal schemes of codimension defined by the maximal minors of a matrix where is a homogeneous polynomial of degree .

In this paper we address the following three fundamental problems: To determine (1) the dimension of (resp. ) in terms of and , (2) whether the closure of is an irreducible component of , and (3) when is generically smooth along . Concerning question (1) we give an upper bound for the dimension of (resp. ) which works for all integers and , and we conjecture that this bound is sharp. The conjecture is proved for , and for under some restriction on and . For questions (2) and (3) we have an affirmative answer for and , and for under certain numerical assumptions.

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D. Bayer and M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra. Source and object code available for Unix and Macintosh computers. Contact the authors, or download from ftp://math.harvard.edu via anonymous ftp.

D. Bayer and M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra. Source and object code available for Unix and Macintosh computers. Contact the authors, or download from ftp://math.harvard.edu via anonymous ftp.